The University of Michigan Combinatorics Seminar


Abstract 

First I will discuss joint work with Alex Postnikov and David Speyer (arXiv:0706.2501) in which we use toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian G(k,n)+. G(k,n)+ is a cell complex whose cells C_G can be parameterized in terms of the combinatorics of certain planar graphs G. To each cell C_G we associate a certain polytope P(G), whose combinatorial structure is reminiscent of the wellknown Birkhoff polytopes: the face lattice of P(G) can be described in terms of matchings and unions of matchings of G. We also demonstrate a connection between the polytopes P(G) and matroid polytopes. We then associate a toric variety to each P(G), and use our technology to prove that the cell decomposition of G(k,n)+ is in fact a CW complex. Next I will briefly describe very recent joint work with Konstanze Rietsch in which we extend the previous work to show that the LusztigRietsch cell decomposition of the totally nonnegative part of any G/P is a CW complex. In the proof, the combinatorics of planar graphs is replaced by the technology of Lusztig's canonical basis. 